From
the 1870s, Americans began to play a larger role in discovering new
mathematical principles. Some studied abroad, especially in Germany, and a
few American universities began to offer graduate degrees in mathematics.

Mathematicians purchased teaching equipment linked to recent advances in
their discipline. They and others also introduced devices to help the
rapidly growing number of high school students.

WELLESLEY
COLLEGE MATH CLASS

From ancient times, mathematicians had studied regular solids like
the tetrahedron, the cube, and the octahedron. In the 1850s, the Swiss
mathematician Ludwig Schläfli described the six regular figures, or
polytopes, that can exist in four-dimensional space.

In 1880, W. I. Stringham, a
fellow at Johns Hopkins University, wrote his Ph.D. dissertation on these
figures. A few years later, the German mathematician and model maker
Ludwig Brill published models of projections of polytopes. These examples
on the right were purchased and displayed by Wesleyan University in Connecticut.

College
students studied figures that could be generated by the motion of straight
lines. Thismodel shows two
such surfaces, the cone on the inside and the hyperboloid on the outside.
Surfaces of this sort had been discovered by the Swiss mathematician
Leonhard Euler (1707-1783), and were useful in engineering
education.

GEOMETRIC
MODEL--ELLIPTIC CONE AND HYPERBOLOID OF ONE SHEET

Americans
also began to build geometric models that reflected recent discoveries.
Richard P.
Baker, an English-born mathematics professor, did advanced work at the
University of Chicago and then taught at the University of Iowa. He
designed and sold several models of surfaces discovered by the German
mathematician G. F. Bernhard Riemann and his students.

GEOMETRIC
MODEL--RIEMANN SURFACE

In
1900, Edgar J. Townsend returned from advanced studies in Germany and
became director of the mathematics department at the University of
Illinois. He promptly arranged for the university to extend its collection
of German geometric models and introduced a graduate program in
mathematics.